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<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:61"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">C</font> is <a href="glib_001.html#V8">Cycle-like</a> &amp; <font color="Maroon">C</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/>

<a NAME="E3:61"/>
<font color="Maroon">C</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:61"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="glib_001.html#K15">.length()</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:61"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="glib_001.html#K15">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 8 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:61"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9
 
<b>by </b><i><a class="ref" href="glib_001.html#T113">GLIB_001:113</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:61"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/>
<a NAME="E9:61"/><i><font color="Green">E59</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<a NAME="E10:61"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">C</font> is <a href="glib_001.html#V1">closed</a>
 
<b>by </b><i><a class="txt" href="chord.html#E15"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> or ( not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>suppose </b></a><a NAME="E1:61_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">i</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:61_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:61_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E5:61_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 is  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>take </b>
<font color="Maroon">i</font>
;<br/><b>take </b>
<font color="Maroon">j</font>
;<br/><a NAME="E7:61_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E8:61_1_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E9:61_1_1"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:61_1_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>thus </b><a NAME="E11:61_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E12:61_1_1"/>
( (  not <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> ) &amp; (  not <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 ) &amp; (  not <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 or  not <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> ) )
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:61_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E8:61"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="chord.html#E9:61"><i><font color="Green">E59</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div>
<b>end;</b></div><b>reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E15:61_1_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 9 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 6</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>then reconsider </b><font color="Maroon">Ci</font> = <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E17:61_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 9 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 8</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E18:61_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_2_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_2_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="glib_001.html#K4">.last()</a> 
 <b>by </b><i/>;<br/><a NAME="E3:61_1_1_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E4:61_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E20:61_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">Ci</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E21:61_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Ci</font>
 <b>by </b><i>, <a class="txt" href="chord.html#E9:61"><i><font color="Green">E59</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E22:61_1_1"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon">Cj</font> = <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="chord.html#E8:61"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_3_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_3_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:61_1_1_3_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i/>;<br/><a NAME="E3:61_1_1_3_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="glib_001.html#K4">.last()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E4:61_1_1_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_3_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_1_1_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E25:61_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">Cj</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:61"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E26:61_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">Cj</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Cj</font>
 ;<br/><a NAME="E2:61_1_1_4"/><b>then </b>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="txt" href="chord.html#E8:61"><i><font color="Green">E56</font></i></a>, , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_4"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E28:61_1_1"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2"><b>suppose </b></a><a NAME="E1:61_1_2"/><i><font color="Green">E81</font></i>: 
( not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:61_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="txt" href="chord.html#E6:61"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:61_1_2"/><b>then </b>
( 1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 )
 <b>by </b><i>, </i>;<br/><a NAME="E4:61_1_2"/><b>then </b>
3 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:61_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:61_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2 is  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 2 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>take </b>
<font color="Maroon">i</font>
;<br/><b>take </b>
<font color="Maroon">j</font>
;<br/><a NAME="E9:61_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E10:61_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 )
 <b>by </b><i><a class="txt" href="chord.html#E8:61"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E11:61_1_2"/>
( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 ;<br/><div><i><font color="Green">E85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_2_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 ;<br/><a NAME="E2:61_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 4
 ;<br/><b>hence </b><a NAME="E3:61_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="chord.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:61_1_2"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_2_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 )
 ;<br/><a NAME="E2:61_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 9 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 3</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_2_3"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 ;<br/><a NAME="E2:61_1_2_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 4
 ;<br/><b>hence </b><a NAME="E3:61_1_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, </i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_2_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:61_1_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E9:61"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="chord.html#T15" target="_self">Th15</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div>
<b>end;</b></div><a NAME="E17:61_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E18:61_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">Ci</font> = <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E20:61_1_2"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="chord.html#T15" target="_self">Th15</a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E21:61_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">Ci</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_2_5"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">Ci</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:61_1_2_5"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_5"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E23:61_1_2"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E24:61_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="txt" href="chord.html#E8:61"><i><font color="Green">E56</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">Cj</font> = <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E26:61_1_2"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:61"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E27:61_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">Cj</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_2_6"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">Cj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:61_1_2_6"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:61_1_2_6"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E29:61_1_2"/>
<font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
